Algebra Examples

$\frac{x}{x - 2} - \frac{3}{x + 2} + \frac{x + 3}{x^{2} - 4}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\frac{x}{x - 2} - \frac{3}{x + 2} + \frac{x + 3}{x^{2} - 2^{2}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{x}{x - 2} - \frac{3}{x + 2} + \frac{x + 3}{(x + 2) (x - 2)}$

Step 2

To write $\frac{x}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{x}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{3}{x + 2} + \frac{x + 3}{(x + 2) (x - 2)}$

Step 3

To write $- \frac{3}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{x}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{3}{x + 2} \cdot \frac{x - 2}{x - 2} + \frac{x + 3}{(x + 2) (x - 2)}$

Step 4

Write each expression with a common denominator of $(x - 2) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x}{x - 2}$ by $\frac{x + 2}{x + 2}$ .

$\frac{x (x + 2)}{(x - 2) (x + 2)} - \frac{3}{x + 2} \cdot \frac{x - 2}{x - 2} + \frac{x + 3}{(x + 2) (x - 2)}$

Step 4.2

Multiply $\frac{3}{x + 2}$ by $\frac{x - 2}{x - 2}$ .

$\frac{x (x + 2)}{(x - 2) (x + 2)} - \frac{3 (x - 2)}{(x + 2) (x - 2)} + \frac{x + 3}{(x + 2) (x - 2)}$

Step 4.3

Reorder the factors of $(x - 2) (x + 2)$ .

$\frac{x (x + 2)}{(x + 2) (x - 2)} - \frac{3 (x - 2)}{(x + 2) (x - 2)} + \frac{x + 3}{(x + 2) (x - 2)}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{x (x + 2) - 3 (x - 2)}{(x + 2) (x - 2)} + \frac{x + 3}{(x + 2) (x - 2)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{x (x + 2) - 3 (x - 2) + x + 3}{(x + 2) (x - 2)}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 - 3 (x - 2) + x + 3}{(x + 2) (x - 2)}$

Step 6.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 2 - 3 (x - 2) + x + 3}{(x + 2) (x - 2)}$

Step 6.3

Move $2$ to the left of $x$ .

$\frac{x^{2} + 2 \cdot x - 3 (x - 2) + x + 3}{(x + 2) (x - 2)}$

Step 6.4

Apply the distributive property.

$\frac{x^{2} + 2 x - 3 x - 3 \cdot - 2 + x + 3}{(x + 2) (x - 2)}$

Step 6.5

Multiply $- 3$ by $- 2$ .

$\frac{x^{2} + 2 x - 3 x + 6 + x + 3}{(x + 2) (x - 2)}$

Step 7

Simplify by adding terms.

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Step 7.1

Subtract $3 x$ from $2 x$ .

$\frac{x^{2} - x + 6 + x + 3}{(x + 2) (x - 2)}$

Step 7.2

Combine the opposite terms in $x^{2} - x + 6 + x + 3$ .

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Step 7.2.1

Add $- x$ and $x$ .

$\frac{x^{2} + 0 + 6 + 3}{(x + 2) (x - 2)}$

Step 7.2.2

Add $x^{2}$ and $0$ .

$\frac{x^{2} + 6 + 3}{(x + 2) (x - 2)}$

Step 7.3

Add $6$ and $3$ .

$\frac{x^{2} + 9}{(x + 2) (x - 2)}$